controllability of fractional stochastic neutral integro-differential equations with state-dependent delay in frechet spaces

This paper investigates the existence and control-lability of state-dependent delay functionally neutral stochastic fractional integro-differential equations within fréchet spaces. by employing fixed-point techniques, the properties of solution operators, and tools from fractional calculus, we derive the conditions under which these systems are controllable. our results extend the theoretical framework of fractional differential equations and provide new insights into the behavior of stochastic systems with memory effects. the findings have broad implications for various scientific fields, particularly those that model systems exhibiting complex dynamics, such as physics, biology, and engineering. to illustrate the practical application of our theoretical results, we provide a detailed example. this research contributes to the deeper understanding of fractional stochastic systems and lays the groundwork for future studies in the area of controlled fractional systems.